Analytical and hybrid solutions of diffusion problems within arbitrarily shaped regions via integral transforms

Abstract

Linear diffusion problems defined within irregular multidimensional regions are analytically solved through integral transforms, requiring numerical routines only for integration purposes, when a general functional boundary representation is considered. Auxiliary one-dimensional eigenvalue problems mapping the irregular region are applied with an integral transformation procedure so that the original differential SturmLiouville system gives place to an algebraic eigenvalue problem. The exact analytical inversion formula is then employed to yield the desired potential, explicitly, at any point within the domain. To allow for improved flexibility and further applicability, the related integration is simplified through an approximate boundary representation using lines connecting user provided points instead of the former exact representation of the irregular bounds, which is particularly advantageous when a functional description of the boundaries is not available. A cylindrical region test case with known exact solution is considered, and treated as an irregular region in the Cartesian coordinates system. Convergence behavior and error analysis are carefully undertaken and illustrated.

Atividades Científicas e Tecnológicas no Campo da Engenharia Mecânica

“…Microcontroladores 42, 44, 55 P Padrão 24,25,26,27,70,71,73 Parafuso Estojo 70,71,72,73,74,75 Perfis Formados a Frio 56, 57, 58, 69…”

“…The authors conclude the work by stating that the solution's convergence via GITT proved to be excellent and completely in agreement with the reference results. They also concluded that automatic error control and computational cost reduction are hallmarks of GITT for this class of problems.Sphaier and Cotta[27] used GITT to obtain solutions of Sturm-Liouville eigenvalue problems, described by multidimensional partial differential models within irregularly shaped domains. Through integral transformations, the successive elimination of independent variables generates a problem of associated algebraic eigenvalue, solved by algorithms from scientific computational libraries.…”

A General Analytical Solution for the Multidimensional Transient Linear Hydraulic Diffusivity Equation in Heterogeneous and Anisotropic Porous Media

“…This methodology uses the General Integral Transform Technique [13][14][15] to handle the nonhomogeneities of the proposed problem, by defining an eigenfunction expansion in terms of a simpler Sturm-Liouville problem of known solution 11-12-22-23 . This methodology can be combined with several analytical methodologies available in the literature 11,12,[17][18][19][20][21] to obtain different solutions for different geometries and different orthogonal coordinate systems. This methodology can be combined with several analytical methodologies available in the literature 11,12,[17][18][19][20][21] to obtain different solutions for different geometries and different orthogonal coordinate systems.…”

“…Once the eigenvalues  i and eigenfunctions ψ( i ,x) are obtained, the parameters N i e f i can be easily computed from (16) and (17) and the potential P h may be evaluated at a given position vector x at a given time t.…”

“…The GITT has been applied to obtain analytical solution of eigenvalue problems for heterogeneous media 16 , irregular domains 17,18 , atmospheric pollutant dispersion 19 , conjugated conduction-convection heat transfer 20 , convection-diffusion problems in petroleum reservoir engineering 21 and several other non-homogeneous diffusion problems. Therefore, GITT is a robust technique that can be applied to obtain analytical solutions for diffusion problems related to reservoir engineering.…”